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A question on consistency in propositional logic.

Mathelogician

Member
Aug 6, 2012
35
Hi everybody!

We have a theorem in natural deduction as follows:
Let H be a set of hypotheses:
====================================
H U {~phi) is inconsistent => H implies (phi).
====================================
Now the question arises:

Let H={p0} for an atom p0. So H U{~p0}={p0 , ~p0}.
We know that {p0 , ~p0} is inconsistent, so by our theorem we would have:
{p0} implies ~p0.
Which we know is impossible.(because for example it means that ~p0 is a semantical consequence of p0).

Now what's wrong here?
Thanks
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Well, your theorem or schema is negating the phi, which you're not doing. In your example, you should end up with {p0} implies p0. No doubt Evgeny can correct any mistakes I just made.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,503
Well, your theorem or schema is negating the phi, which you're not doing. In your example, you should end up with {p0} implies p0.
You are right. If we apply the theorem to H U{~p0}, then phi from the theorem is p0. Therefore, the theorem concludes that {p0} implies p0.
 

Mathelogician

Member
Aug 6, 2012
35
Oooooooops!